Food & Process Engineering Technology textbook

298 Pages·6.43 MB·english

by  Wilhelm L.R.

ERRATA (February 2005)
Food & Process Engineering Technology textbook
L.R. Wilhelm, D.A. Suter, G.H. Brusewitz
ISBN: 1-892769-43-3

Abstract: Food & Process Engineering Technology errata (February 2005)

Keywords: Food & Process Engineering Technology, errata, erratum, textbook

Page 19 Problem 1.1. Item j. should be 9.008 × 104

Page 27 Figure 2.01. The title under the upper left image should be Roundness = A /A ,
p c
and the title under the lower left image should be Roundness Ratio = r/R.

Page 52 Problem 2.30.a., The temperature should be 70°C (not 20°C).
2 2
P ⎛ N ⎞ ⎛D ⎞ ρ
Page 97 Equation 4.27 should be t1 =⎜ 1⎟ ⎜ 1⎟ 1 (rho instead of P).
⎜ ⎟ ⎜ ⎟
Pt2 ⎝N2⎠ ⎝D2⎠ ρ2
Page 108 Problem 4.1. Units for the velocity squared term should be ft2/s2.

Page 139 Problem 5.9. The sheathing is medium density particleboard.

Page 161 Problem 6.3, All times should be in minutes (not seconds).

Page 204 Line 4, the equation should be:
FT = 262 500 kJ/m3 [0.000 100 m3 K/W + 0.000 008 9 m3 K/W]/20 K

Page 235 Figure 9.05. h = 335 kJ/kg (not 3.35).
sf

Page 301 Table A.8, Units for Heat of Respiration are mW/kg.

Suggested improvement items
Page 51 Suggested Alternate Problems
2.4.1. Name two fruit/vegetables that are good examples of: (a) having nearly round
shape and (b) having elliptical shape.
2.5. Compute the roundness of a square object having sides 1.0 cm long.
2.7. Compute the sphericity of a cube having 1.0 cm sides.
Introduction:
Problem-Solving Tools
Abstract. This chapter provides a general introduction to common mathematical
concepts used in engineering applications. Simple algebraic applications, including
exponents and logarithms, are reviewed. Other useful topics include: interpolation,
graphical presentations, the rate equation, and the concept of mass and energy
balances.
Keywords. Equations, graphs, interpolation of tabular data, logarithms, mass
balances, rate equation, significant digits, spreadsheets.
1.1 Introduction
Knowledge of basic problem-solving principles is essential for understanding many
of the engineering applications in food engineering. This unit includes a review of
several basic problem-solving topics needed to understand subjects covered in
subsequent units. Guidelines for presentation of information are also included. The
material is presented as a review only. If you understand the topics, you do not need to
read this unit. If you do not understand them, you should read the unit, study the
examples, and work the problems given at the end of the unit.
1.2 Significant Digits
The writing of any number indicates a certain degree of precision for that number.
This is best shown by the use of examples as shown below.
Table 1.01. Significant digit examples.
Number Tolerance Significant Digits
2.2 2.2 ± 0.05 2
2.20 2.20 ± 0.005 3
3.1416 3.1416 ± 0.00005 5
2 2 ± 0.5 1
0.0030 0.0030 ± 0.00005 2
2200 2200 ± 50 2
2200 2200 ± 5 3
2200 2200 ± 0.5 4
Note the significance of the zeros in the examples above. The numbers 2.2 and 2.20
represent different degrees of precision because the added zero is a significant digit.
On the other hand, zeros following a decimal but preceding the first non-zero digit are
2 Food & Process Engineering Technology

not significant digits since they only serve to place the decimal. The last number
(2200) is ambiguous since it may have been rounded to the nearest whole number (4
significant digits) or the nearest 10 or the nearest 100. This ambiguity can be elimi-
nated by the use of scientific notation (e.g., 2.200 × 103 and 2.2 × 103 show four and
two significant digits respectively.)
The precision of results from any arithmetic calculation depends upon the precision
of the least precise number used in the calculations. Thus if two numbers are multi-
plied (i.e., 2.2 × 3.1416) the result, 6.9115, implies false precision because it contains
more significant digits than the least accurate factor in the multiplication. Thus the
result in this example should be rounded to 6.9. This procedure applies identically for
division. For addition and subtraction, the results should be rounded to the number of
decimal points included in the least precise number.
The above comments represent the official guidelines for significant digits. In
actual practice, these rules are not always followed. For example, a length may be
expressed as 1 m when it is actually 1.0 m or 1.00 m. Because numbers are often
presented in less than the true number of significant digits, it is common practice to
present calculated results as 3 significant digits—even though the rules may call for
only one or two significant digits.
You should also be cautious in rounding numbers. A series of numerical calcula-
tions should be carried out with several significant digits and the final answer rounded
to an appropriate number of significant digits. For example, 10/3 equals 3.33333…,
not 3 or 3.3.
1.3 Unit Factors
Numerical quantities without units generally carry little meaning in engineering
problems. Consequently, including the correct units with the numerical answer to a
problem is just as important as arriving at the correct numerical value. A brief
discussion of units and a list of many common unit conversion factors are included in
the appendix.
Unit analysis affords a valuable aid in solving many physical problems. Some
problems require only the conversion of units for a solution. Others may require a
more thorough analysis of the problem before finding the number and units for the
solution. Virtually all problems involving food engineering applications involve
numbers with units. Thus, close monitoring of units is essential.
The conversion of units is accomplished by the use of unit factors, defined as
ratios whose actual value is unity, or one. Consider the following:
Equation Ratio Unit Factor
1 hr 1hr
1 hr = 3600 s =1
3600 s 3600s

3600s 3600s
1 hr = 3600 s =1
1hr 1hr
1000 L 1000L
1 m3 = 1000 L = 1
1 m 3 1m3
Chapter 1 Introduction: Problem-Solving Tools 3
The combination of numbers and units in the numerator of a unit factor is equal to
the combination of numbers and units in the denominator. Thus, the actual value of the
ratio is one, or unity. The numerical values alone are usually not equal to one since
they serve as conversion multipliers for the units involved.
The following example shows how a unit factor problem can be set up for a sys-
tematic solution. The steps may seem unnecessary for this simple problem; however,
you can avoid difficulties with more complicated problems by following this proce-
dure.

Example 1.1

A conveyor belt moves 3 ft in 15 s. What is the belt speed in meters per hour?
Solution:
First enter a blank followed by the final desired units. An equality sign should
follow this, and then enter the pertinent information needed for the solution:
m 3 ft
=
hr 15s
Once the pertinent information is included, the solution consists of multiplying
the basic data on the right hand side of the equation by appropriate unit factors
to obtain the desired final results. Many different unit factors can usually be
used to obtain the desired results; however, the better solutions will be those that
produce a logical solution with a minimum number of unit factors. The follow-
ing is one such “better” solution:
m 3 ft 1 m 3600 s
= × × =219.4
hr 15 s 3.281 ft 1 hr
The units on the right side of the equation are canceled to produce the resulting
units on the left side of the equation. The numeric answer is then written in the
blank on the left side of the equation.

1.4 Algebraic Equations
An equation is a statement of equality between one or more expressions involving
variables and constants. The equation for a straight line (y = ax + b) is a simple
example. We solve equations by manipulating them such that the equality of the
equation is not affected. Such changes are: addition or subtraction of the same number
or variable to each side (y – y = ax + b – y and b + y = ax + b + b); multiplication of
each side by the same number or variable (ky = kax + kb); or dividing each side by the
same non-zero number or variable (y/a = x + b/a).
4 Food & Process Engineering Technology

Algebraic equations are used extensively in analysis of food processing operations.
To adequately use such equations, one must:
1. Be able to perform simple algebraic manipulations that change the form of the
equation;
2. Be able to interpret graphical representations of these equations; and
3. Be able to solve the equations for unknown parameters.
Introductory coverage of these topics is presented in this and subsequent units.
Several geometric formulas commonly used in food engineering applications are
shown below. More extensive lists of formulas are available in physics and mathemat-
ics texts, mathematical handbooks, and other similar sources.
πD2
Circle A= =πr2 C=πD
4
4 πD3
Sphere A=πD2 V = πr3=
3 6
Cylinder A=2πrh=πDh V =πr2h
where: A = area
D = diameter
C = circumference
r = radius
V = volume
1.5 Exponents and Logarithms
Relationships involving roots, exponents, and logarithms are common in food
engineering applications. Since roots are actually exponents ( 2 =21/2), we will
examine briefly the rules for exponents and logarithms. An understanding of these
rules is essential to understanding the governing equations applicable to many food-
engineering analyses. Food sterilization and drying processes are two major applica-
tions that involve logarithmic (and thus exponential) applications in their analyses.
1.5.1 Exponents
In many mathematical operations we must raise a constant or variable to a power.
This power may be an integer or decimal value, and it may be positive or negative. We
use exponents to identify these powers. In the relationships given below a may be a
constant or a variable while m and n are integer or decimal exponents.
a −n = 1 1
a n n a =an
a m
am ×an =am+n = a m−n
a n
(am)n =amn n am =amn
Chapter 1 Introduction: Problem-Solving Tools 5
1.5.2 Logarithms
The logarithm of X to the base b is the exponent to which b must be raised to get X. Thus:
102 = 100 ⇒ log (100) = 2
10
log (10) = 1 ⇒ 101 = 10
10
log (3.162) = 0.5 ⇒ 100.5 = 3.162.
10
We commonly use two bases for logarithms: common logarithms, or logarithms
to the base 10, and natural logarithms, logarithms to the base e. The number e (2.718
281 828 459 ...) is a constant that occurs frequently in mathematical problems. If a
subscript is not given, log X usually means log X, and ln X means log X.
10 e
Three simple rules involving logarithms are given in the equations below. These
rules are often very useful in working with logarithmic equations.
log XY =log X +logY
X
log =logX −logY
Y
log Xn = nlog X
Most logarithmic calculations are now performed using hand calculators or com-
puters. Calculations can also be made manually using logarithmic tables. Table 1.02 is
an abbreviated table of natural logarithms (base e). Common logarithms (base 10) are
given in Table 1.03. Typical tables of common logarithms do not show that the
numbers in the table header row are decimal. In typical tables, these numbers are
written as whole numbers 0 – 9 rather than 0.00 – 0.9. In addition, all numbers in the
table are assumed to be decimal values and the “1.” is not shown. Table 1.03 is read
by finding the intersection of the whole number and the decimal value. Thus, log
10
12.6 = 1.1004. Hand-held calculators are commonly used for problem solving. Most
now include logarithmic functions; however, the specific computational procedure
differs among calculators. One useful application of the logarithmic tables is to verify

Table 1.02. Natural (base e; e = 2.71828) logarithm of selected numbers.
n logn n logn n logn n logn
e e e e
0.0 2.0 0.6931 11 2.3979 40 3.6889
0.1 -2.3026 2.5 0.9163 12 2.4849 50 3.9120
0.2 -1.6094 3.0 1.0986 13 2.5649 60 4.0943
0.3 -1.2040 3.5 1.2528 14 2.6391 70 4.2485
0.4 -0.9163 4.0 1.3863 15 2.7081 80 4.3820
0.5 -0.6931 4.5 1.5041 16 2.7726 90 4.4998
0.6 -0.5108 5.0 1.6094 17 2.8332 100 4.6052
0.7 -0.3567 6.0 1.7918 18 2.8904 200 5.2983
0.8 -0.2231 7.0 1.9459 19 2.9444 300 5.7038
0.9 -0.1054 8.0 2.0794 20 2.9957 400 5.9915
1.0 0.0000 9.0 2.1972 25 3.2189 500 6.2146
1.5 0.4055 10.0 2.3026 30 3.4012 1000 6.9078
6 Food & Process Engineering Technology

Table 1.03. Four-place common (base 10) logarithms.
N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 -- -- -- -- -- -- -- -- --
1.0000 0.6990 0.5229 0.3979 0.3010 0.2218 0.1549 0.0969 0.0458
1 0 0.0414 0.0792 0.1139 0.1461 0.1761 0.2041 0.2304 0.2553 0.2788
2 0.3010 0.3222 0.3424 0.3617 0.3802 0.3979 0.4150 0.4314 0.4472 0.4624
3 0.4771 0.4914 0.5051 0.5185 0.5315 0.5441 0.5563 0.5682 0.5798 0.5911
4 0.6021 0.6128 0.6232 0.6335 0.6435 0.6532 0.6628 0.6721 0.6812 0.6902
5 0.6990 0.7076 0.7160 0.7243 0.7324 0.7404 0.7482 0.7559 0.7634 0.7709
6 0.7782 0.7853 0.7924 0.7993 0.8062 0.8129 0.8195 0.8261 0.8325 0.8388
7 0.8451 0.8513 0.8573 0.8633 0.8692 0.8751 0.8808 0.8865 0.8921 0.8976
8 0.9031 0.9085 0.9138 0.9191 0.9243 0.9294 0.9345 0.9395 0.9445 0.9494
9 0.9542 0.9590 0.9638 0.9685 0.9731 0.9777 0.9823 0.9868 0.9912 0.9956
10 1.0000 1.0043 1.0086 1.0128 1.0170 1.0212 1.0253 1.0294 1.0334 1.0374
20 1.3010 1.3032 1.3054 1.3075 1.3096 1.3118 1.3139 1.3160 1.3181 1.3201
30 1.4771 1.4786 1.4800 1.4814 1.4829 1.4843 1.4857 1.4871 1.4886 1.4900
40 1.6021 1.6031 1.6042 1.6053 1.6064 1.6075 1.6085 1.6096 1.6107 1.6117
50 1.6990 1.6998 1.7007 1.7016 1.7024 1.7033 1.7042 1.7050 1.7059 1.7067
60 1.7782 1.7789 1.7796 1.7803 1.7810 1.7818 1.7825 1.7832 1.7839 1.7846
70 1.8451 1.8457 1.8463 1.8470 1.8476 1.8482 1.8488 1.8494 1.8500 1.8506
80 1.9031 1.9036 1.9042 1.9047 1.9053 1.9058 1.9063 1.9069 1.9074 1.9079
90 1.9542 1.9547 1.9552 1.9557 1.9562 1.9566 1.9571 1.9576 1.9581 1.9586
100 2.0000 2.0004 2.0009 2.0013 2.0017 2.0022 2.0026 2.0030 2.0035 2.0039

Table 1.04. Logarithms of selected numbers.
ln 0.1 = –2.3026 log 0.1 = –1.00
10
ln 0.5 = –0.6931 log 1 = 0.000
10
ln 1.0 = 0.000 log 50 = 1.6990
10
ln 50 = 3.9120 log 50.5 = 1.7033
10
ln 100 = 4.6052 log 43576 = 4.6392
10
ln 1000 = ln (10 × 100) = ln 10 + ln 100
= 2.3026 + 4.6052 = 6.9078
Chapter 1 Introduction: Problem-Solving Tools 7
Table 1.05. Calculation of logarithms.
Problem Logarithmic Calculations Answer
5521.1 ⇒ln(55)21.1 = ln(55) = 4.0073=1.9083 ⇒ e1.9083 6.74
2.1 2.1
5521.1 ⇒log(55)21.1 = log(55) =1.7404 =0.8287 ⇒ 100.8287 6.74
2.1 2.1
(3.5)3 ⇒ln(3.5)3 =3ln(3.5)=3(1.2528)=3.758 ⇒ e3.758 42.9
35
⇒log(3.5)3=3log(3.5)=3log =3(log(35)−log(10)
(3.5)3 10 ⇒ 101.632 42.9
=3(1.5441−1)=3(0.5441)=1.632
correct use of your calculator. Table 1.04 shows examples of logarithmic table values
that could be used for such verification. Examples in Table 1.05 demonstrate the use
of logarithms to solve other problems.
1.6 Graphs and Coordinate Systems
Graphs are used to present data in visual form and to show relationships between
equation variables. We are often interested in obtaining an equation to relate experi-
mental data. A straight line can represent many such experimental data if plotted on an
appropriate graph. Table 1.06 summarizes three types of graphs, the equation type,
and examples of each. Many different coordinate systems are used in the scientific and
business world. Most graphs, however, are presented as standard rectilinear X-Y
(Cartesian), semi-logarithmic (semi-log), or full logarithmic (log-log) graphs. These

Table 1.06. Straight-line relationships for common graph types.
Graph Type Rectilinear Semi-log Log-log
Equation Type Linear Exponential power law
General Form of y = be mx
y = mx + b y = bx m
Equation y = b10 mx
Intercept [a] b = y – mx -- --
lny −lny
m= 2 1
y − y x −x logy −logy
Slope [b] m= 2 1 2 1 m= 2 1
x −x logy −logy logx −logx
2 1 or m= 2 1 2 1
x −x
2 1
[a] Intercept has meaning only for rectilinear coordinates. The value of y is never equal to zero
for semilog and log-log graphs.
[b] A common way of stating the slope for semi-log graphs is the change in x corresponding to
a full log cycle on the y-axis.
8 Food & Process Engineering Technology

Figure 1.01. Examples of linear, semi-log, and log-log graphs.
(Image from a Microsoft Excel worksheet.)
are the only three types of graphs we will consider. Figure 1.01 shows examples of
these three graphs. Actual curves plotted are:
Linear: y=0.5x+10
Semi-log: y=0.5×100.23x
Log-log: y=0.5x0.21
1.6.1 General (Cartesian) Coordinates
This type of graph uses equal increments for all measurements along an axis. The
scales along the vertical and horizontal axes may be different but the grid (lines along
each axis) usually has uniform spacing. Figure 1.02 is an example of a graph in
Cartesian coordinates. This type of graph, commonly called an X-Y graph, is perhaps
the most common graph type for presenting engineering data. The curve shapes in this
figure indicate that the data could also be printed on a semi-logarithmic graph.
1.6.2 Logarithmic Graphs
Logarithmic graphs present data along an axis that is scaled by taking the loga-
rithms (base 10) of the numbers to be plotted. For example: log 1 = 0; log 2 = 0.301;
log 4 = 0.602; log 9 = 0.954; log 10 = 1.00; log 20 = 1.301; log 40 = 1.602; and log
100 = 2.00. We can see from these values that each multiple of 10 begins a new cycle
on a logarithmic scale.
Chapter 1 Introduction: Problem-Solving Tools 9
1.0
No Recirculation
0.9
30% Recirculation
0.8 60% Recirculation
ht 90% Recirculation
g 0.7
ei
W
al 0.6
niti
of I 0.5
n
o 0.4
rti
o
p 0.3
o
r
P
0.2
0.1
0.0
0 10 20 30 40 50
Time (hours)

Figure 1.02. Moisture removal as a function of time for peaches
dried with recirculated air.
Logarithmic graphs may use logarithmic scales on both axes (called log-log) or on
only one axis with a standard scale on the other (called semi-log). We will be con-
cerned only with semi-log graphs—a graph type used extensively in thermal process-
ing analysis. An example of a semi-logarithmic graph is shown in Figure 1.03.
1.6.3 Plotting Graphs
Certain general rules should be followed when plotting all graphs. The first rule is
that a graph should clearly present the desired information. This requires that appro-
priate X and Y values must be shown on the axes. In addition each axis should have a

100
Theory
Data
s
nit
U
r of 10
e
b
m
u
N
1
0 20 40 60 80 100
Time (minutes)
Figure 1.03. Example of a semi-logarithmic plot.